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Register a new userGeneralizations of Alladis formula for arithmetical semigroups
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In this article, we prove that a general version of Alladis formula with Dirichlet convolution holds for arithmetical semigroups satisfying Axiom $A$ or Axiom $A^{#}$. As applications, we apply our main results to certain semigroups coming from algebraic number theory, arithmetical geometry and graph theory, particularly generalizing the results of Wang 2021, Kural et al. 2020 and Duan et al. 2020.
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Using Bochner-Martinelli type residual currents we prove some generalizations of Jacobis Residue Formula, which allow proper polynomial maps to have `common zeroes at infinity, in projective or toric situations.
We give a new formula for the Chebotarev densities of Frobenius elements in Galois groups. This formula is given in terms of smallest prime factors $p_{mathrm{min}}(n)$ of integers $ngeq2$. More precisely, let $C$ be a conjugacy class of the Galois group of some finite Galois extension $K$ of $mathbb{Q}$. Then we prove that $$-lim_{Xrightarrowinfty}sum_{substack{2leq nleq X[1pt]left[frac{K/mathbb{Q}}{p_{mathrm{min}}(n)}right]=C}}frac{mu(n)}{n}=frac{#C}{#G}.$$ This theorem is a generalization of a result of Alladi from 1977 that asserts that largest prime divisors $p_{mathrm{max}}(n)$ are equidistributed in arithmetic progressions modulo an integer $k$, which occurs when $K$ is a cyclotomic field $mathbb{Q}(zeta_k)$.
In this paper some generalizations of the sum of powers of natural numbers is considered. In particular, the class of sums whose generating function is the power of the generating function for the classical sums of powers is studying. The so-called binomial sums are also considered. The problem of constructing polynomials that allow to calculate the values of the corresponding sums in certain cases is solved.
We extend the notions of quasi-monomial groups and almost monomial groups, in the framework of supercharacter theories, and we study their connection with Artins conjecture regarding the holomorphy of Artin $L$-functions.
In this paper we prove an explicit formula for the arithmetic intersection number (CM(K).G1)_{ell} on the Siegel moduli space of abelian surfaces, generalizing the work of Bruinier-Yang and Yang. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for the intersection number (CM(K).G1)_{ell} under strong assumptions on the ramification of the primitive quartic CM field K. Yang later proved this conjecture assuming that O_K is freely generated by one element over the ring of integers of the real quadratic subfield. In this paper, we prove a formula for (CM(K).G1)_{ell} for more general primitive quartic CM fields, and we use a different method of proof than Yang. We prove a tight bound on this intersection number which holds for all primitive quartic CM fields. As a consequence, we obtain a formula for a multiple of the denominators of the Igusa class polynomials for an arbitrary primitive quartic CM field. Our proof entails studying the Embedding Problem posed by Goren and Lauter and counting solutions using our previous article that generalized work of Gross-Zagier and Dorman to arbitrary discriminants.
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